Rectified 9-orthoplexes

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.
There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the tetrahedral cell centers of the 9-orthoplex.
These polytopes are part of a family 511 uniform 9-polytopes with BC₉ symmetry.

Rectified 9-orthoplex

The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb.

Alternate names

rectified enneacross

Construction

There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C₉ or Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D₉ or Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length are all permutations of:

Root vectors

Its 144 vertices represent the root vectors of the simple Lie group D₉. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B₉ and C₉ simple Lie groups.

Rectified 9-orthoplexes